Suarez claims the club agreed to let him leave Anfield if Liverpool failed to qualify for this season's Champions League.
The club refute that, and while Carragher claims the Uruguayan is "too good" for both Liverpool and Arsenal, and better suited to the likes of Real Madrid, Barcelona and Bayern Munich, he expects greater loyalty from the striker.
Carragher said in his Daily Mail column: "It didn't surprise me that Brendan Rodgers has made him train on his own. I never thought Luis would down tools in the way Fernando Torres did when he wanted to leave Liverpool - his attitude over the final 18 months became progressively worse - but now Suarez has given me reason to think again.
"To see it come to this is very dispiriting. A lot of good people have stood by Luis since he joined the club. Rodgers and Kenny Dalglish could not have done more for him. In the darkest times, we wore T-shirts to show we were standing alongside him, even if it was ill-advised.
"So it is laughable to hear he is unhappy with Liverpool for not letting him join Arsenal."
Suarez was banned by the Football Association for racially abusing Manchester United defender Patrice Evra, and again towards the end of last season for biting Chelsea's Branislav Ivanovic.
Despite those offences, Liverpool still want him at the club, however Carragher claims he has seen at close quarters a change in Suarez's attitude to life at Liverpool.
He said: "I took part in a training session at Melwood before Steven Gerrard's testimonial last Saturday and I was alarmed by the Luis Suarez with whom I was sharing a pitch.
"From his first day at Liverpool, I had been used to seeing a warrior who loved his football and charged around with great enthusiasm.
"On this occasion, however, his attitude was completely different. He didn't want the ball and had no interest in getting involved. He stood on the periphery, with shoulders slumped."
While Carragher spent his entire career with Liverpool, before retiring in May, Michael Owen moved away to join Real Madrid in 2004.
And Owen, who also retired at the end of last season, can understand why Suarez, at the age of 26, would want to leave and join a club playing at the highest level in Europe.
Owen told the Daily Telegraph: "People have this stereotypical view of a footballer that he's just greedy, no loyalty, but you can bet your bottom dollar that if Suarez wasn't as good was he was then Liverpool wouldn't have stood by him [during his misdemeanours]. Of course, you have to bear in mind the things people have done for you along the way, and have some gratitude, but you have one shot.
"Luis has not come through the ranks. He doesn't have a huge affinity with Liverpool but the fans have a huge affinity with him and expect that to be reciprocated. But he wants to go and be at the best club he can.
"This is where I can't quite understand it. If it's Arsenal the only option for him, I can't really understand why he would want to leave given the way the Liverpool fans feel about him and the team is virtually built about him. If Real Madrid came in for him, I could certainly understand that."